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Best of Donald Catlin
I recently received an email from one of my readers, Amy, posing a question about Blackjack. Here is Amy's email:
First of all Amy, thank you for writing to me. Your email brings up some interesting points. To illustrate them I would like to critique your email and then pose the question that I think you should be asking.
In a one-dimensional random walk (which is what a gambler's stake is), the probability of success is the probability of ruin subtracted from 1. Since you postulate an unlimited bankroll the probability of ruin is zero hence one would conclude that success is certain. However, since you are playing a negative game it is certainly possible that one could lose more and more as play continues and never reach the desired goal. In other words, postulating an unlimited bankroll can lead to an erroneous conclusion.
My second criticism is in the sentence fragment "do I have more chance of walking away a winner every time." More chance than what? This is too vague and I don't know how to answer this.
Amy I think you want to know what your chances of success are when starting with a large stake and playing for modest gains. For example, let's say you start with $1000 and, making $50 bets, play to win $100. If you go back to my October 2011 article I derive a formula for approximating Blackjack with a one-dimensional random walk with a fixed step size (a fixed wager). Here are the numbers you need.
We might as well start with 20 units and play for 2, it's the same situation. So using the notation in my article:
S= 1.1225, M = -0.0058, B = 20, G = 2, and w = 1
Using these numbers the ruin probability turned out to be 0.144778471 so the probability of success is 0.855221528. You can adjust B to be a larger number but make sure that your calculator has accuracy to many decimal places. Best wishes Amy and good luck playing Blackjack. See you all next month.
Don Catlin can be reached at firstname.lastname@example.org
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